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For the critical focusing wave equation  in the radial case, we establish the role of the “center stable” manifold  constructed in Krieger and Schlag (Am J Math 129(3):843–913, ) near the ground state (, 0) as a threshold between blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń et al. (Nonlinearity 17(6):2187–2201, ). The underlying topology is stronger than the energy norm.

Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions